A180000 -id:A180000 - cp's OEIS frontend

A048619 a(n) = LCM(binomial(n,0), ..., binomial(n,n)) / binomial(n,floor(n/2)).

1, 1, 1, 1, 2, 1, 3, 3, 4, 2, 10, 5, 30, 15, 7, 7, 56, 28, 252, 126, 60, 30, 330, 165, 396, 198, 286, 143, 2002, 1001, 15015, 15015, 7280, 3640, 1768, 884, 15912, 7956, 3876, 1938, 38760, 19380, 406980, 203490, 99484, 49742, 1144066, 572033, 1961256, 980628

Offset: 0

Labos Elemer

			If n=10 then A002944(10)=2520, A001405(10)=252, the quotient a(10)=10.

Index entries for sequences related to lcm's

Cf. A001405, A002944.

[Lcm([1..n+1]) div (Floor((n+3)/2)*Binomial(n+1,Floor((n+3)/2))): n in [0..50]]; // Vincenzo Librandi, Jul 10 2019

Table[Apply[LCM, Binomial[n, Range[0, n]]]/Binomial[n, Floor[n/2]], {n, 0, 48}] (* Michael De Vlieger, Jun 29 2017 *)

{A048619(n) = lcm(vector(n+1, i, i)) / binomial(n+1, (n+1)\2) / ((n+2)\2);}

a(n) = A002944(n)/A001405(n).
a(n) = lcm(1..n+1)/(floor((n+3)/2)*binomial(n+1,floor((n+3)/2))). - Paul Barry, Jul 03 2006
a(n) = lcm(1,2,...,n+1) / (ceiling((n+1)/2)*binomial(n+1,floor((n+1)/2))) = A003418(n+1) / A100071(n+1). - Max Alekseyev, Oct 23 2015
a(n) = A263673(n+1) / A110654(n+1) = A180000(n+1) / A152271(n). - Max Alekseyev, Oct 23 2015
a(2*n-1) = A068553(n) = A068550(n)/n.

Definition corrected and a(0)=1 prepended by Max Alekseyev, Oct 23 2015

A068550 a(n) = lcm{1, ..., 2n} / binomial(2n, n).

Original entry on oeis.org

1, 1, 2, 3, 12, 10, 30, 105, 56, 252, 1260, 330, 1980, 2574, 2002, 15015, 240240, 61880, 15912, 151164, 38760, 406980, 4476780, 1144066, 13728792, 24515700, 6249100, 84362850, 21474180, 5462730, 81940950, 1270084725, 645122400

Offset: 0

N. J. A. Sloane, Mar 23 2002

Known to be always an integer.

Hojoo Lee, Re: LCM [1,2,..,N] > 2^{N-1}, NMBRTHRY Mailing List, Feb 18 2002.
Daniel Ropp, Solution to Problem 31, Crux Mathematicorum, Vol. 13, No. 10 (1987), p. 314.

Bisection of A180000 and A263673.

a[0] = 1; a[n_] := (LCM @@ Range[2*n])/Binomial[2*n, n]; Array[a, 33, 0] (* Amiram Eldar, Mar 06 2022 *)

a(n) = lcm([1..2*n])/binomial(2*n, n); \\ Michel Marcus, Mar 06 2022

a(n) = A099996(n) / A000984(n) = A003418(2*n) / A001405(2*n) = A180000(2*n) = A263673(2*n).

a(n) = n * A068553(n) = n * A048619(2*n-1).

a(0)=1 prepended by Max Alekseyev, Oct 23 2015

A163076 Primes of the form k$ - 1. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

5, 19, 29, 139, 251, 12011, 48619, 51479, 155117519, 81676217699, 1378465288199, 5651707681619, 386971244197199, 1580132580471899, 30067266499541039, 6637553085023755473070799, 35257120210449712895193719, 399608854866744452032002440111

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			Since 4$ = 6 the prime 5 is listed.

Jinyuan Wang, Table of n, a(n) for n = 1..53
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A055490, A056040, A163078 (arguments k), A163074, A163075.

a := proc(n) select(isprime, map(x -> A056040(x)-1,[$1..n])); sort(%) end:

Reap[Do[f = n!/Quotient[n, 2]!^2; If[PrimeQ[p = f - 1], Sow[p]], {n, 1, 70}]][[2, 1]] // Union (* Jean-François Alcover, Jun 28 2013 *)

More terms from Jinyuan Wang, Mar 22 2020

A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

2, 3, 5, 31, 67, 139, 631, 9743, 16253, 17977, 27901, 37589

Offset: 1

Peter Luschny, Jul 21 2009

a(n) are the primes in A163077.

			5 is prime and 5$ + 1 = 30 + 1 = 31 is prime, so 5 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A062363, A093804, A163077.

Cf. A056040. - Robert G. Wilson v, Aug 09 2010

a := proc(n) select(isprime,select(k -> isprime(A056040(k)+1),[$0..n])) end:

f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; p = 2; lst = {}; While[p < 38000, a = f@p + 1; If[ PrimeQ@a, AppendTo[ lst, p]; Print@p]; p = NextPrime@p]; lst (* Robert G. Wilson v, Aug 08 2010 *)

is(k) = isprime(k) && ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(8)-a(12) from Robert G. Wilson v, Aug 08 2010

A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 9, 13, 12, 2, 19, 2, 5, 36, 6, 19, 43, 11, 47, 67, 39, 41, 70, 12, 17, 83, 88, 81, 25, 53, 91, 97, 106, 79, 43, 39, 7, 29, 73, 6, 79, 115

Offset: 1

Peter Luschny, Jul 24 2009

nonn

If this is zero, p is a swinging Wilson prime.

			The swinging Wilson quotient related to the 5th prime is (252+1)/11=23, so the 5th term is 23 mod 11 = 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A163211, A002068, A163210.

WR := proc(f,r,n) map(p->(f(p-1)+r(p))/p mod p,select(isprime,[$1..n])) end:
A002068 := n -> WR(factorial,p->1,n);
A163213 := n -> WR(swing,p->(-1)^iquo(p+2,2),n);

sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; Mod[(sf[p - 1] + (-1)^Floor[(p + 2)/2])/p, p]); Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Jun 28 2013 *)

sf(n)=n!/(n\2)!^2
apply(p->sf(p-1)\/p%p, primes(100)) \\ Charles R Greathouse IV, Dec 11 2016

A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, -9, -7, -4, 0, 6, 44, 35, 28, 24, 24, 30, -165, -121, -86, -58, -34, -10, 20, 594, 429, 308, 222, 164, 130, 120, 140, -2037, -1443, -1014, -706, -484, -320, -190, -70, 70, 6824, 4787, 3344, 2330, 1624, 1140, 820, 630, 560, 630

Offset: 0

Peter Luschny, Aug 05 2009

An analog to the derangement triangle (A068106).

			1
0, 1
1, 1, 2
2, 3, 4, 6
-9, -7, -4, 0, 6
44, 35, 28, 24, 24, 30
-165, -121, -86, -58, -34, -10, 20

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Row sums are A163773.

Cf. A056040, A163650, A163771, A163772, A068106.

DiffTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] - A[j] od;
for i from 0 to m do T := T,(-1)^(m-i)*A[i] od;
if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL,[T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
Computes n rows of the triangle.
A163770 := n -> DiffTria(k->swing(k),n,true);
A068106 := n -> DiffTria(k->factorial(k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[(-1)^(n - i)*Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040).

A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 16, 11, 8, 6, 47, 31, 20, 12, 6, 146, 99, 68, 48, 36, 30, 447, 301, 202, 134, 86, 50, 20, 1380, 933, 632, 430, 296, 210, 160, 140, 4251, 2871, 1938, 1306, 876, 580, 370, 210, 70, 13102, 8851, 5980, 4042, 2736, 1860, 1280, 910, 700, 630

Offset: 0

Peter Luschny, Aug 06 2009

Triangle read by rows.

An analog to the binomial triangle of the factorials (A076571).

			Triangle begins
    1;
    2,   1;
    5,   3,   2;
   16,  11,   8,   6;
   47,  31,  20,  12,  6;
  146,  99,  68,  48, 36, 30;
  447, 301, 202, 134, 86, 50, 20;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Row sums are A163843.

Cf. A056040, A163865, A163841, A163842, A163650.

SumTria := proc(f,n,display) local m,A,j,i,T; T:=f(0);
for m from 0 by 1 to n-1 do A[m] := f(m);
for j from m by -1 to 1 do A[j-1] := A[j-1] + A[j] od;
for i from 0 to m do T := T,A[i] od;
if display then print(seq(T[i],i=nops([T])-m..nops([T]))) fi;
od; subsop(1=NULL,[T]) end:
swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
# Computes n rows of the triangle:
A163840 := n -> SumTria(swing,n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[i], {i, k, n}]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

T(n,k) = Sum_{i=k..n} binomial(n-k,n-i)*i$ where i$ denotes the swinging factorial of i (A056040), for n >= 0, k >= 0.

A163841 Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).

Original entry on oeis.org

1, 3, 2, 11, 8, 6, 45, 34, 26, 20, 195, 150, 116, 90, 70, 873, 678, 528, 412, 322, 252, 3989, 3116, 2438, 1910, 1498, 1176, 924, 18483, 14494, 11378, 8940, 7030, 5532, 4356, 3432, 86515, 68032, 53538

Offset: 0

Peter Luschny, Aug 06 2009

For n >= 0, k >= 0 let T(n,k) = sum{i=k..n} binomial(n-k,n-i)*(2i)$ where i$ denotes the swinging factorial of i (A056040). Triangle read by rows.

			Triangle begins
     1;
     3,    2;
    11,    8,    6;
    45,   34,   26,   20;
   195,  150,  116,   90,   70;
   873,  678,  528,  412,  322,  252;
  3989, 3116, 2438, 1910, 1498, 1176,  924;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Row sums are A163844. Cf. A056040, A163650, A163841, A163842, A163840, A026375, A002426, A000984.

Computes n rows of the triangle. For the functions 'SumTria' and 'swing' see A163840.
a := n -> SumTria(k->swing(2*k),n,true);

sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[Binomial[n - k, n - i]*sf[2*i], {i, k, n}]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)

A163077 Numbers k such that k$ + 1 is prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 14, 15, 24, 27, 31, 38, 44, 45, 49, 67, 76, 92, 99, 119, 124, 133, 136, 139, 144, 168, 171, 185, 265, 291, 332, 368, 428, 501, 631, 680, 689, 696, 765, 789, 890, 1034, 1233, 1384, 1517, 1615, 1634, 1809, 2632, 2762, 3925, 4419, 5108, 5426

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			0$ + 1 = 1 + 1 = 2 is prime, so 0 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002981, A163078, A163079, A163080.

a := proc(n) select(x -> isprime(A056040(x)+1),[$0..n]) end:

fQ[n_] := PrimeQ[1 + 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]]; Select[ Range[0, 8660], fQ] (* Robert G. Wilson v, Aug 09 2010 *)

is(k) = ispseudoprime(1+k!/(k\2)!^2); \\ Jinyuan Wang, Mar 22 2020

a(45)-a(56) from Robert G. Wilson v, Aug 09 2010

A163078 Numbers k such that k$ - 1 is prime. Here '$' denotes the swinging factorial function (A056040).

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 13, 15, 18, 30, 35, 39, 41, 47, 49, 58, 83, 86, 102, 111, 137, 151, 195, 205, 226, 229, 317, 319, 321, 368, 389, 426, 444, 477, 534, 558, 567, 738, 804, 882, 1063, 1173, 1199, 1206, 1315, 1624, 1678, 1804, 2371, 2507, 2541, 2844, 3084, 3291

Offset: 1

Peter Luschny, Jul 21 2009

nonn

			4$ - 1 = 6 - 1 = 5 is prime, so 4 is in the sequence.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Primes.

Cf. A002982, A056040, A163077, A163079, A163080.

a := proc(n) select(x -> isprime(A056040(x)-1),[$0..n]) end:

fQ[n_] := PrimeQ[ -1 + 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]]; Select[ Range@ 3647, fQ] (* Robert G. Wilson v, Aug 09 2010 *)

is(k) = ispseudoprime(k!/(k\2)!^2-1); \\ Jinyuan Wang, Mar 22 2020

a(42)-a(54) from Robert G. Wilson v, Aug 09 2010

cp's OEIS Frontend

A048619 a(n) = LCM(binomial(n,0), ..., binomial(n,n)) / binomial(n,floor(n/2)).

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A068550 a(n) = lcm{1, ..., 2n} / binomial(2n, n).

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A163076 Primes of the form k$ - 1. Here '$' denotes the swinging factorial function (A056040).

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A163079 Primes p such that p$ + 1 is also prime. Here '$' denotes the swinging factorial function (A056040).

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A163213 Swinging Wilson remainders ((p-1)$ + (-1)^floor((p+2)/2))/p mod p, p prime. Here '$' denotes the swinging factorial function (A056040).

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A163770 Triangle read by rows interpolating the swinging subfactorial (A163650) with the swinging factorial (A056040).

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Maple

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A163840 Triangle interpolating the binomial transform of the swinging factorial (A163865) with the swinging factorial (A056040).

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